How to Save Particle Combing Data to a Normal Map Image File in UV Space?
I do not have an answer for you, although I can provide you with a few nuggets of info that will come in handy.

Here is some python code that shows you how to dump the coordinates of the root and tip of the hairs:

import bpy

def dump_particles(ps):
    for p in ps.particles:
        print(p.location)
        #print(p.fuv)
        print(p.hair_keys[-1].co_local)
        print("_")

dump_particles(bpy.context.active_object.particle_systems[0])


The major stumbling block is that the barycentric coordinate of the hair is NOT exposed via the python API.  (see also How do I comb hair in python?)

This makes calculating the position of the hair in UV coordinates much harder (although you could probably reverse-engineer it from the location if you are willing to do a lot of analytic geometry).

Even once you do figure out where the hair is in UV space, you'll need to choose how to interpolate to compute the values for the image you want to use as a normal map.  (Maybe compute the dual of a voronoi diagram and use barycentric coordinates again).